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\begin{document}

\title{Complex Analysis}
\subtitle{Chapter 7. Elliptic Functions}
%\institute{SLUC}
\author{LVA}
%\date
%\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{ {2023年9月21日} }

\maketitle

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\begin{frame}{Contents 1-2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}
\item Simply Periodic Functions

\begin{enumerate}
\item[1.1.] Representation by Exponentials
\item[1.2.] The Fourier Development
\item[1.3.] Functions of Finite Order
\end{enumerate}

\item 
\begin{enumerate}
\item Doubly Periodic Functions
\item[2.1.] The Period Module
\item[2.2.] Unimodular Transformations
\item[2.3.] The Canonical Basis
\item[2.4.] General Properties of Elliptic Functions
\end{enumerate}

\end{enumerate}

\end{frame}

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\begin{frame}{Contents 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{enumerate}
\item[3.] The Weierstrass Theory

\begin{enumerate}
\item[3.1.] The Weierstrass p-function
\item[3.2.] The Functions t(z) and u(z)
\item[3.3.] The Differential Equation
\item[3.4.] The Modular Function A(r)
\item[3.5.] The Conformal Mapping by A(r)
\end{enumerate}
 
\end{enumerate}

\end{frame}

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\begin{frame}{1.1. Representation by Exponentials. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The simplest function with period $\omega$ is the exponential $e^{2\pi i z/\omega}$. It is a fundamental fact that any function with period $\omega$ can be expressed in terms of this particular function.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.2. The Fourier Development. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Assume that $\Omega'$ contains an annulus $r_1 < |\zeta| < r_2$ in which $F$ has no poles. In this annulus $F$ has a Laurent development
$$
F(\zeta) = \sum\limits_{n=-\infty}^{\infty} c_n\zeta^n,
$$
and we obtain
$$
f(z) = \sum\limits_{n=-\infty}^{\infty} c_n e^{2\pi i nz/\omega}. 
$$
This is the complex Fourier development of $f(z)$, valid in the parallel strip
that corresponds to the given annulus. 
}

\item  Answer. 
\begin{enumerate}
\item 
 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{1.3. Functions of Finite Order. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. }

\item  Answer. 
\begin{enumerate}
\item When $\Omega$ is the whole plane $F(\zeta)$ has isolated singularities at $\zeta = 0$ and $\zeta = \infty$.
If both these singularities are inessential, that is, either removable singularities or poles, then $F$ is a rational function. We say in this case that $f$ has finite order, equal to the order of $F$.
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.1. The Period Module. Theorem 1. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
A discrete module consists either of zero alone, of the integral multiples $nw$ of a single complex number $w\neq 0$, or of all linear combinations $n_1w_1+n_2w_2$ with  integral coefficients of two numbers $w_1$, $w_2$ with nonreal ratio $w_2/w_1$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.2. Unimodular Transformations. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. }

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.3. The Canonical Basis. Theorem 2. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
There exists a basis $(w_1,w_2)$ such that the ratio $r = w_2/w_1$ satisfies the following conditions: (i) $\mathrm{Im} \tau > 0$, (ii) $-\frac{1}{2} < \mathrm{Re} \tau \le \frac{1}{2}$, (iii) $|\tau| \ge 1$, (iv) $\mathrm{Re} \tau \ge 0$ if $|\tau| = 1$. The ratio $\tau$ is uniquely determined by these conditions, and there is a choice of two, four, or six corresponding bases.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.4. General Properties of Elliptic Functions. Theorem 3. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
An elliptic function without poles is a constant. 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.4. General Properties of Elliptic Functions. Theorem 4.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The sum of the residues of an rllliptic function is zero.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.4. General Properties of Elliptic Functions. Theorem 5.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
A nonconstant elliptic function has equally many poles as it has zeros.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{2.4. General Properties of Elliptic Functions. Theorem 6.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The zeros $a_1, \cdots, a_n$ and poles $b_1,\cdots, b_n$ of an elliptic function satisfy $a_1 + \cdots +a_n \cong b_1 + \cdots + b_n (\mod M)$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.1. The Weierstrass $\mathcal{P}$-function. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. }

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.2. The Functions $\zeta(z)$ and $\sigma(z)$. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. }

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.2. The Functions $\zeta(z)$ and $\sigma(z)$. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that any even elliptic function with periods $w_1$, $w_2$ can be expressed in the form
$$
C\prod\limits_{k=1}^{n} \frac{\mathcal{P}(z)-\mathcal{P}(a_k)}{\mathcal{P}(z)-\mathcal{P}(b_k)} \hspace{0.5cm} (C = \mathrm{const}.)
$$
provided that $0$ is neither a zero nor a pole. What is the corresponding form if the function either vanishes or becomes infinite at the origin? 
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.2. The Functions $\zeta(z)$ and $\sigma(z)$. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that any elliptic function with periods $w_1$, $w_2$ can be written as
$$
C\prod\limits_{k=1}^{n} \frac{\sigma(z-a_k)}{\sigma(z-b_k)} \hspace{0.5cm} (C = \mathrm{const}.)
$$
Hint: Use (14) and Theorem 6.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Differential Equation.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
$$
\mathcal{P}'(z)^2 = 4\mathcal{P}^3 - g_2\mathcal{P}-g_3. 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Differential Equation. Exercise - 1}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Use (14) to show that the right-hand member is a periodic function of $z$.
Find the multiplicative constant by comparing the Laurent developments.
$$\mathcal{P}(z) - \mathcal{P}(u) = - \frac{\sigma(z-u)\sigma(z+u)}{\sigma(z)^2\sigma(u)^2}. $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Differential Equation. Exercise - 2}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Follows from (16) by taking logarithmic derivatives.
$$\frac{\mathcal{P}'(z)}{\mathcal{P}(z) - \mathcal{P}(u)} 
= \zeta(z-u) + \zeta(z+u) -2\zeta(z). $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Differential Equation. Exercise - 3}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
This is a symmetrized version of (17).
$$
\zeta(z+u) = \zeta(z) + \zeta(u) + \frac{1}{2}
\frac{\mathcal{P}'(z)-\mathcal{P}'(u)}{\mathcal{P}(z) - \mathcal{P}(u)}. $$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Differential Equation. Exercise - 4}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The addition theorem for the g;J-function
$$
\mathcal{P}(z+u) = - \mathcal{P}(z) - \mathcal{P}(u) 
+\frac{1}{4}\left( 
\frac{\mathcal{P}'(z)-\mathcal{P}'(u)}{\mathcal{P}(z) - \mathcal{P}(u)}
\right)^2. 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Differential Equation. Exercise - 5}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
$$
\mathcal{P}(2z) = \frac{1}{4}\left(\frac{\mathcal{P}''(z)}{\mathcal{P}'(z)}\right)^2
-2\mathcal{P}(z).
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Differential Equation. Exercise - 6}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
$$
\mathcal{P}'(z) = - \frac{\sigma(2z)}{\sigma(z)^4}. 
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.3. The Differential Equation. Exercise - 7}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
$$
\begin{vmatrix} 
\mathcal{P}(z) & \mathcal{P}'(z) & 1 \\ 
\mathcal{P}(u) & \mathcal{P}'(u) & 1 \\ 
\mathcal{P}(u+z) & -\mathcal{P}'(u+z) & 1 \\ 
\end{vmatrix} =0.
$$
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{3.4. The Modular Function $\lambda(\tau)$.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. }

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{The Conformal Mapping by $\lambda(\tau)$. Theorem 7. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
The modular function $\lambda(\tau)$ effects a one-to-one conformal mapping of the region $\Omega$ onto the upper half plane. The mapping extends continuously to the boundary in such a way that $\tau = 0, 1, \infty$ correspond to $\lambda = 1, \infty, 0$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{The Conformal Mapping by $\lambda(\tau)$. Theorem 8.}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Every point $\tau$ in the upper half plane is equivalent under the congruence subgroup $\mod 2$ to exactly one point inn $\bar{\Omega}\cup\Omega'$.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{The Conformal Mapping by $\lambda(\tau)$. Exercise 1. }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}

\begin{itemize}
\item  {\color{red}Question. 
Show that the function
$$
J(\tau) = \frac{4}{27}\frac{(1-\lambda+\lambda^2)^3}{\lambda^2(1-\lambda)^2}
$$
is automorphic with respect to the full modular group. 
Where does it take the values 0 and 1, and with what multiplicities? 
Show that
$$
J(\tau) = \frac{-4(e_1e_2 + e_2e_3 + e_3e_1)^3} 
{(e_1-e_2)^2(e_2-e_3)^2(e_3-e_1)^2}.
$$
Show also that $J(\tau)$ maps the region $\Delta$ in Fig. 7-4 onto a half plane.
}

\item  Answer. 
\begin{enumerate}
\item 
 
\item 

\item 

\end{enumerate}

\end{itemize}

\end{frame}

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\end{document}

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